9 THE CAUCHY FORMULA 225

By assumption, there is in A a loop homotopy <p, defined in I x J, of
y into a circuit reduced to a point. As x $ (p(l x J), Cauchy's theorem shows

that f dz/(z - x) = 0.
Jy

9. THE CAUCHY FORMULA

(9.9.1) Let A <= C be a simply connected domain (Section 9.7), f an analytic
mapping of A into a complex Banach space E. For any circuit y in A, defined in I
and any x e A — y(I), we have (Cauchy's formula)

7(* ;?)/(*) =T-:

2niJy z — x

Consider the function g(z) defined in A, equal to (/(z) —f(x))/(z — x)
for z ^ x, to f'(x) at the point x; g is analytic in A, for it is
obviously analytic in A—{x} by (9.3.2) and (9.5.1); on the other
hand, there is a ball B c A of center x such that for z e B, /(z) =
f(x) + (z - x)f(x) + • • • + (z - x)nf(n\x)/n\ + • • • the series being conver-
gent in B; this proves that for any z e B, g(z) is equal to the sum of the
convergent series

hence analytic at x. From Cauchy's theorem (9.6.3) we have | g(z) dz = 0,

Jy
and writing

z — x z — x

yields (9.9.1) by definition of the index.

Conversely:

(9.9.2) Let y be a road in C, defined in an interval I = [6, c] and let g be a
continuous mapping ofy(l) into a complex Banach space E. Then

rfx) dx

is defined and analytic in the complement of y(I); more precisely, for any
point a 6 C — y(I), if we write

the power series (cn(z — a)n) is convergent in any open ball B of center a con-
tained in C — y(I), and its sum is equal tof(z) in B. y(^ 5 ?) = 0. But the exterior of D is connected, as